Abstract
We present a procedure which allows us to recover classical and nonclassical logical structures as concrete logics associated with physical theories expressed by means of classical languages. This procedure consists in choosing, for a given theory \({{\mathcal{T}}}\) and classical language \({{\fancyscript{L}}}\) expressing \({{\mathcal{T}}, }\) an observative sublanguage L of \({{\fancyscript{L}}}\) with a notion of truth as correspondence, introducing in L a derived and theory-dependent notion of C-truth (true with certainty), defining a physical preorder \(\prec\) induced by C-truth, and finally selecting a set of sentences ϕ V that are verifiable (or testable) according to \({{\mathcal{T}}, }\) on which a weak complementation ⊥ is induced by \({{\mathcal{T}}. }\) The triple \((\phi_{V},\prec,^{\perp})\) is then the desired concrete logic. By applying this procedure we recover a classical logic and a standard quantum logic as concrete logics associated with classical and quantum mechanics, respectively. The latter result is obtained in a purely formal way, but it can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Hence quantum logic can be considered as a mathematical structure formalizing the properties of the notion of verification in quantum physics. This conclusion supports the general idea that some nonclassical logics can coexist without conflicting with classical logic (global pluralism) because they formalize metalinguistic notions that do not coincide with the notion of truth as correspondence but are not alternative to it either.
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Notes
Of course, truth and C-truth are different notions in our approach. But it must be noted that the identification of true with certain, or certainly true, is basic in some approaches to the foundations of QM (in particular, in the Geneva–Brussels approach (Piron 1976; Aerts 1999)). We show in Sect. 4 that true and certainly true coincide in CM if pure states only are considered, which may lead one to overlook the deep difference between the two notions.
A measurement can be described as an interaction between a physical object and a measuring apparatus in CM. In a real measurement the apparatus is in a mixed state because one never knows all its properties at a microscopic level, hence probabilities must be introduced in the theoretical description (which admit an ignorance interpretation, hence are epistemic). Unsharp properties and contextuality then occur if one wants to refer to the physical object only, avoiding a complete description of the interaction with the measuring apparatus (the “hidden measurement” processes in the case of Aerts’ quantum machine). It must be noted, however, that a deeper form of nonlocal contextuality occurs in QM according to an orthodox view (Bell 1964; Greenberger et al. 1990; Mermin 1993) and that this kind of contextuality is avoided in the ESR model mentioned in Sect. 1 (Garola and Sozzo 2009, 2010, 2011a, b).
Note that a similar objection does not occur in the case of predicates denoting states, because the extension ext(S) of a state \({S \in {\fancyscript{S}}}\) can be interpreted as the set of all physical objects that are actually prepared in the state S.
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The authors are greatly indebted with Prof. Carlo Dalla Pozza and Dr. Marco Persano for reading the manuscript and providing useful remarks and suggestions.
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Garola, C., Sozzo, S. Recovering Quantum Logic Within an Extended Classical Framework. Erkenn 78, 399–419 (2013). https://doi.org/10.1007/s10670-011-9353-4
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DOI: https://doi.org/10.1007/s10670-011-9353-4